Geometric Sequence

From Math Images

Revision as of 15:48, 28 June 2012 by Pzhao1 (Talk | contribs)
(diff) ←Older revision | Current revision (diff) | Newer revision→ (diff)
Jump to: navigation, search
This is a Helper Page for:
Logarithmic Spirals
Koch Snowflake
Taylor Series

A geometric sequence is a sequence whose terms change by a constant factor.

A few examples of sequences are below:

(1)5,25,125,625,3125\,

(2)81,-27,9,-3,1,-\frac{1}{3}

It should be apparent that the two sequences are increasing by 5 and -\frac{1}{3}, respectively. As a result, we can determine the value of any given term in the sequence if we know its position.

We examine sequence (1) to find this formula

We identify the first term, 5, as a_0 and the following terms 25, 125, 625,... \, as a_1, a_2, a_3,...\,.

We observe

a_1 = a_0 * 5\,

a_2 = a_1 * 5 = a_0 * 5^2\,

a_3 = a_2 * 5 = a_0 * 5^3\,

a_4 = a_3 * 5 = a_0 * 5^4\,

From here we can reason that a_n = a_0 * 5^n\,.

We can generalize further that the constant factor of change does not need to be 5 but actually can be any value r.

Thus we have the expression

a_n = a_0 * r^n\,


At this point, we must make the distinction between the words series and sequence. A sequence is a list of numbers of finite or infinite length; for example, (1) is geometric sequence of length 5. A series is a sum of numbers of a sequence.

Other Properties

Another property of geometric series is that any term is the square root of the product of its two neighbors. For example:

\sqrt{5*125} = \sqrt{5^4} = 5^2\ =25,

In fact, so long as the absolute difference between the two values to be square rooted is the same integer, the property will hold.

a_n = a_0 * 5^n = \sqrt{(a_0 * 5^{(n-d)})*(a_0 * 5^{(n+d)})}

Summing the Sequence

A very convenient formula for summing a geometric sequence can found from algebraic manipulation.

s = a + ar + ar^2 + ar^3 + \cdots

We divide both sides by a,

(1) \frac{s}{a} = 1 + r + r^2 + r^3 + \cdots

Here we multiply both sides by r,

(2) \frac{rs}{a} = r + r^2 + r^3 + r^4 + \cdots

We now subtract equation (2) from (1), which results in

\frac{s - rs}{a} = 1

Simplifying for s gives us,

s = \frac{a}{1-r}

This gives us the formula for an infinite geometric series.


Similarly, we use a similar process to find the sum of a finite geometric series.

s = a + ar + ar^2 + ar^3 + \cdots + ar^n

We divide both sides by a,

(3)\frac{s}{a} = 1 + r + r^2 + r^3 + \cdots + r^n

Here we multiply both sides by r,

(4)\frac{rs}{a} = r + r^2 + r^3 + r^4 + \cdots + r^{n+1}

We now subtract equation (4) from (3), which results in

\frac{s - rs}{a} = 1 - r^{n+1}

Simplifying for s gives us,


s = \frac{a*(1 - r^{n+1})}{1-r}

This gives us the formula for an finite geometric series.

Interactive Demonstration

If you can see this message, you do not have the Java software required to view the applet.

Personal tools